Answer:
e^(3z) = xyz
to find ∂z/∂x → consider z as a function of x and take y to be a constant … but be careful when you do it b/c it’s easy to mess up
so differentiating with respect to x:
e^(3z) * 3 * ∂z/∂x = z * y + xy * 1 * ∂z/∂x … [using the chain rule on the LHS and the product rule on the RHS]
Factor out the ∂z/∂x:
∂z/∂x [3e^(3z) – xy] = yz
∂z/∂x = yz / [3e^(3z) – xy]
Do the same thing to find ∂z/∂y except consider z to be a function of y and take x to be a constant …
Differentiating with respect to y:
e^(3z) * 3 * ∂z/∂y = z * x + xy * 1 * ∂z/∂y
∂z/∂y [3e^(3z) – xy] = xz
∂z/∂y = xz / [3e^(3z) – xy]